Patterns in Rational Base Number Systems

Number systems with a rational number a/b>1 as base have gained interest in recent years. In particular, relations to Mahler’s $\frac{3}{2}$ -problem as well as the Josephus problem have been established. In the present paper we show that the patterns of digits in the representations of positive integers in such a number system are uniformly distributed. We study the sum-of-digits function of number systems with rational base a/b and use representations w.r.t. this base to construct normal numbers in base a in the spirit of Champernowne. The main challenge in our proofs comes from the fact that the language of the representations of integers in these number systems is not context-free. The intricacy of this language makes it impossible to prove our results along classical lines. In particular, we use self-affine tiles that are defined in certain subrings of the adèle ring $\mathbb{A}_{\mathbb{Q}}$ and Fourier analysis in $\mathbb{A}_{\mathbb{Q}}$ . With help of these tools we are able to reformulate our results as estimation problems for character sums.     

荧光寿命的快速傅里叶变换拟合方法

介绍了一种利用快速傅里叶变换算法对稀土掺杂物质的荧光寿命进行数据拟合的方法。稀土掺杂物质可用来制备多种光学传感器,用于温度、pH值等多种参量测量领域。本方法利用快速傅里叶变换(FFT)结果作为基础,从非零项的相位角的正切值得出被测的荧光寿命,具有速度快、误差小、不受本底干扰等一系列优点。以掺铒光纤为例,通过数字仿真将本方法与其它几种传统的拟合方法进行了比较。快速傅里叶变换方法的测量偏差不到Prony方法的50％,为对数似合(log-fit)方法测量偏差的1／6。另外,快速傅里叶变换方法由于不受本底噪声影响,可以不必在信号处理时去掉本底噪声,因而可以明显缩短测量时间。     

Generalized Thue-Morse sequences of squares

We consider compact group generalizations T(n) of the Thue-Morse sequence and prove that the subsequence T(n ²) is uniformly distributed with respect to a measure gv that is absolutely continuous with respect to the Haar measure. The proof is based on a proper generalization of the Fourier based method of Mauduit and Rivat in their study of the sum-of-digits function of squares to group representations.